Considering the imperfection of RIS, we assume that the phase noise at RIS can be written as \(\hat{\theta }_n \in \mathcal {U}[-k_r\pi ,k_r\pi ]\)  [1]}, where \(\mathcal {U}\)  denotes the uniform distribution and \(k_r\)  measures the severity of the residual impairments at the RIS. Therefore, the phase shift matrix of the RIS with phase
noise can be expressed as
